The guys who wrote Amazing Spider-Man 2, Alex Kurtzman, Roberto Orci, and Jeff Pinkner, have been rehired to take on The Amazing Spider-Man 3. It's not really a surprising bit of news, as these are the guys that know the story they are telling better than anyone. I imagine they'll write The Amazing Spider-Man 4 as well.
Sony Pictures has already set release dates for the next two movies, which include June 10th, 2016, and May 4th, 2018. Andrew Garfield will continue to play Peter Parker/Spider-Man in the movies, and I imagine Marc Webb will continue to direct them as well.
I'm excited for the next Spider-Man movie. It's going to be fun to see where they take the story, and what other characters and villains that they introduce to the film franchise. There's a lot of Sinister Six talk, which would be great!